Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B

Percentage Accurate: 85.4% → 99.6%

Time: 12.9s

Alternatives: 13

Speedup: 1.9×

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) + (z * log((1.0 - y)))) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * log(y)) + (z * log((1.0d0 - y)))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x * Math.log(y)) + (z * Math.log((1.0 - y)))) - t;
}

Python

def code(x, y, z, t):
	return ((x * math.log(y)) + (z * math.log((1.0 - y)))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * log(y)) + Float64(z * log(Float64(1.0 - y)))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * log(y)) + (z * log((1.0 - y)))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Initial Program: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) + (z * log((1.0 - y)))) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * log(y)) + (z * log((1.0d0 - y)))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x * Math.log(y)) + (z * Math.log((1.0 - y)))) - t;
}

Python

def code(x, y, z, t):
	return ((x * math.log(y)) + (z * math.log((1.0 - y)))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * log(y)) + Float64(z * log(Float64(1.0 - y)))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * log(y)) + (z * log((1.0 - y)))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Alternative 1: 99.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right), y, -1\right) \cdot y, z, -t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma
  (log y)
  x
  (fma (* (fma (fma -0.3333333333333333 y -0.5) y -1.0) y) z (- t))))

C

double code(double x, double y, double z, double t) {
	return fma(log(y), x, fma((fma(fma(-0.3333333333333333, y, -0.5), y, -1.0) * y), z, -t));
}

Julia

function code(x, y, z, t)
	return fma(log(y), x, fma(Float64(fma(fma(-0.3333333333333333, y, -0.5), y, -1.0) * y), z, Float64(-t)))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Log[y], $MachinePrecision] * x + N[(N[(N[(N[(-0.3333333333333333 * y + -0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y), $MachinePrecision] * z + (-t)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.4%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)}\right) - t \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

   2. lower-*.f64 N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

   3. sub-neg N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \left(\color{blue}{\left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot y\right)\right) - t \]

   4. *-commutative N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \left(\left(\color{blue}{\left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) \cdot y} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot y\right)\right) - t \]

   5. metadata-eval N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \left(\left(\left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) \cdot y + \color{blue}{-1}\right) \cdot y\right)\right) - t \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{-1}{3} \cdot y - \frac{1}{2}, y, -1\right)} \cdot y\right)\right) - t \]

   7. sub-neg N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{-1}{3} \cdot y + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, y, -1\right) \cdot y\right)\right) - t \]

   8. metadata-eval N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{fma}\left(\frac{-1}{3} \cdot y + \color{blue}{\frac{-1}{2}}, y, -1\right) \cdot y\right)\right) - t \]

   9. lower-fma.f64 99.8

      \[\leadsto \left(x \cdot \log y + z \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right)}, y, -1\right) \cdot y\right)\right) - t \]

5. Applied rewrites 99.8%

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right), y, -1\right) \cdot y\right)}\right) - t \]
6. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot \log y + z \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, y, \frac{-1}{2}\right), y, -1\right) \cdot y\right)\right) - t} \]

   2. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot \log y + z \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, y, \frac{-1}{2}\right), y, -1\right) \cdot y\right)\right)} - t \]

   3. associate--l+ N/A

      \[\leadsto \color{blue}{x \cdot \log y + \left(z \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, y, \frac{-1}{2}\right), y, -1\right) \cdot y\right) - t\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \log y} + \left(z \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, y, \frac{-1}{2}\right), y, -1\right) \cdot y\right) - t\right) \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\log y \cdot x} + \left(z \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, y, \frac{-1}{2}\right), y, -1\right) \cdot y\right) - t\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, z \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, y, \frac{-1}{2}\right), y, -1\right) \cdot y\right) - t\right)} \]

   7. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{z \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, y, \frac{-1}{2}\right), y, -1\right) \cdot y\right) + \left(\mathsf{neg}\left(t\right)\right)}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{z \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, y, \frac{-1}{2}\right), y, -1\right) \cdot y\right)} + \left(\mathsf{neg}\left(t\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, y, \frac{-1}{2}\right), y, -1\right) \cdot y\right) \cdot z} + \left(\mathsf{neg}\left(t\right)\right)\right) \]

   10. lift-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, y, \frac{-1}{2}\right), y, -1\right) \cdot y\right) \cdot z + \color{blue}{\left(-t\right)}\right) \]

   11. lower-fma.f64 99.8

       \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right), y, -1\right) \cdot y, z, -t\right)}\right) \]

7. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right), y, -1\right) \cdot y, z, -t\right)\right)} \]
8. Add Preprocessing

Alternative 2: 99.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot y, z, \mathsf{fma}\left(\log y, x, -t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma (* (fma -0.5 y -1.0) y) z (fma (log y) x (- t))))

C

double code(double x, double y, double z, double t) {
	return fma((fma(-0.5, y, -1.0) * y), z, fma(log(y), x, -t));
}

Julia

function code(x, y, z, t)
	return fma(Float64(fma(-0.5, y, -1.0) * y), z, fma(log(y), x, Float64(-t)))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(-0.5 * y + -1.0), $MachinePrecision] * y), $MachinePrecision] * z + N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.4%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t} \]

   2. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right)} - t \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(z \cdot \log \left(1 - y\right) + x \cdot \log y\right)} - t \]

   4. associate--l+ N/A

      \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right) + \left(x \cdot \log y - t\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right)} + \left(x \cdot \log y - t\right) \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\log \left(1 - y\right) \cdot z} + \left(x \cdot \log y - t\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(1 - y\right), z, x \cdot \log y - t\right)} \]

   8. lift-log.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\log \left(1 - y\right)}, z, x \cdot \log y - t\right) \]

   9. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\log \color{blue}{\left(1 - y\right)}, z, x \cdot \log y - t\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)}, z, x \cdot \log y - t\right) \]

   11. lower-log1p.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)}, z, x \cdot \log y - t\right) \]

   12. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{-y}\right), z, x \cdot \log y - t\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{x \cdot \log y + \left(\mathsf{neg}\left(t\right)\right)}\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{x \cdot \log y} + \left(\mathsf{neg}\left(t\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{\log y \cdot x} + \left(\mathsf{neg}\left(t\right)\right)\right) \]

   16. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{\mathsf{fma}\left(\log y, x, \mathsf{neg}\left(t\right)\right)}\right) \]

   17. lower-neg.f64 99.8

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, \color{blue}{-t}\right)\right) \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)} \]

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot \left(\frac{-1}{2} \cdot y - 1\right)}, z, \mathsf{fma}\left(\log y, x, -t\right)\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{2} \cdot y - 1\right) \cdot y}, z, \mathsf{fma}\left(\log y, x, -t\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{2} \cdot y - 1\right) \cdot y}, z, \mathsf{fma}\left(\log y, x, -t\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{2} \cdot y + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot y, z, \mathsf{fma}\left(\log y, x, -t\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{-1}{2} \cdot y + \color{blue}{-1}\right) \cdot y, z, \mathsf{fma}\left(\log y, x, -t\right)\right) \]

   5. lower-fma.f64 99.8

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.5, y, -1\right)} \cdot y, z, \mathsf{fma}\left(\log y, x, -t\right)\right) \]

7. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.5, y, -1\right) \cdot y}, z, \mathsf{fma}\left(\log y, x, -t\right)\right) \]
8. Add Preprocessing

Alternative 3: 99.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\log y, x, \left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot z\right) \cdot y\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (fma (log y) x (* (* (fma -0.5 y -1.0) z) y)) t))

C

double code(double x, double y, double z, double t) {
	return fma(log(y), x, ((fma(-0.5, y, -1.0) * z) * y)) - t;
}

Julia

function code(x, y, z, t)
	return Float64(fma(log(y), x, Float64(Float64(fma(-0.5, y, -1.0) * z) * y)) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[Log[y], $MachinePrecision] * x + N[(N[(N[(-0.5 * y + -1.0), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 86.4%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(x \cdot \log y + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right)} - t \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\color{blue}{\log y \cdot x} + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

   2. remove-double-neg N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)} \cdot x + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

   3. mul-1-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \log y}\right)\right) \cdot x + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

   4. mul-1-neg N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(-1 \cdot \log y\right)\right)} \cdot x + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

   5. mul-1-neg N/A

      \[\leadsto \left(\left(-1 \cdot \color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}\right) \cdot x + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

   6. log-rec N/A

      \[\leadsto \left(\left(-1 \cdot \color{blue}{\log \left(\frac{1}{y}\right)}\right) \cdot x + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \log \left(\frac{1}{y}\right), x, y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right)} - t \]

   8. log-rec N/A

      \[\leadsto \mathsf{fma}\left(-1 \cdot \color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}, x, y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(-1 \cdot \color{blue}{\left(-1 \cdot \log y\right)}, x, y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

   10. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(-1 \cdot \log y\right)}, x, y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

   11. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}\right), x, y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

   12. remove-double-neg N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, x, y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

   13. lower-log.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, x, y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right) \cdot y}\right) - t \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \left(-1 \cdot z + \frac{-1}{2} \cdot \color{blue}{\left(z \cdot y\right)}\right) \cdot y\right) - t \]

   16. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \left(-1 \cdot z + \color{blue}{\left(\frac{-1}{2} \cdot z\right) \cdot y}\right) \cdot y\right) - t \]

   17. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(-1 \cdot z + \left(\frac{-1}{2} \cdot z\right) \cdot y\right) \cdot y}\right) - t \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot z\right) \cdot y\right)} - t \]
6. Add Preprocessing

Alternative 4: 89.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.05 \cdot 10^{-127}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot y, z, -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (log y) x (- t))))
   (if (<= x -2.9e-6)
     t_1
     (if (<= x 3.05e-127) (fma (* (fma -0.5 y -1.0) y) z (- t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(log(y), x, -t);
	double tmp;
	if (x <= -2.9e-6) {
		tmp = t_1;
	} else if (x <= 3.05e-127) {
		tmp = fma((fma(-0.5, y, -1.0) * y), z, -t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(log(y), x, Float64(-t))
	tmp = 0.0
	if (x <= -2.9e-6)
		tmp = t_1;
	elseif (x <= 3.05e-127)
		tmp = fma(Float64(fma(-0.5, y, -1.0) * y), z, Float64(-t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision]}, If[LessEqual[x, -2.9e-6], t$95$1, If[LessEqual[x, 3.05e-127], N[(N[(N[(-0.5 * y + -1.0), $MachinePrecision] * y), $MachinePrecision] * z + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.9000000000000002e-6 or 3.0499999999999999e-127 < x

   1. Initial program 95.8%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x \cdot \log y - t} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{x \cdot \log y + \left(\mathsf{neg}\left(t\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot x} + \left(\mathsf{neg}\left(t\right)\right) \]

      3. remove-double-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)} \cdot x + \left(\mathsf{neg}\left(t\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \log y}\right)\right) \cdot x + \left(\mathsf{neg}\left(t\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot \log y\right)\right)} \cdot x + \left(\mathsf{neg}\left(t\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}\right) \cdot x + \left(\mathsf{neg}\left(t\right)\right) \]

      7. log-rec N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\log \left(\frac{1}{y}\right)}\right) \cdot x + \left(\mathsf{neg}\left(t\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \log \left(\frac{1}{y}\right), x, \mathsf{neg}\left(t\right)\right)} \]

      9. log-rec N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}, x, \mathsf{neg}\left(t\right)\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-1 \cdot \color{blue}{\left(-1 \cdot \log y\right)}, x, \mathsf{neg}\left(t\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(-1 \cdot \log y\right)}, x, \mathsf{neg}\left(t\right)\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}\right), x, \mathsf{neg}\left(t\right)\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, x, \mathsf{neg}\left(t\right)\right) \]

      14. lower-log.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, x, \mathsf{neg}\left(t\right)\right) \]

      15. lower-neg.f64 94.7

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{-t}\right) \]

   5. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, -t\right)} \]

   if -2.9000000000000002e-6 < x < 3.0499999999999999e-127

   1. Initial program 69.0%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right) - t} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right) + \left(\mathsf{neg}\left(t\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log \left(1 - y\right) \cdot z} + \left(\mathsf{neg}\left(t\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(1 - y\right), z, \mathsf{neg}\left(t\right)\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)}, z, \mathsf{neg}\left(t\right)\right) \]

      5. lower-log1p.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)}, z, \mathsf{neg}\left(t\right)\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{-y}\right), z, \mathsf{neg}\left(t\right)\right) \]

      7. lower-neg.f64 94.4

         \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{-t}\right) \]

   5. Applied rewrites 94.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{-1}{2} \cdot y - 1\right), z, -t\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 94.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot y, z, -t\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 78.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -4.8 \cdot 10^{+65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right), y, -1\right) \cdot y, z, -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -4.8e+65)
     t_1
     (if (<= x 2.8e+102)
       (fma (* (fma (fma -0.3333333333333333 y -0.5) y -1.0) y) z (- t))
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -4.8e+65) {
		tmp = t_1;
	} else if (x <= 2.8e+102) {
		tmp = fma((fma(fma(-0.3333333333333333, y, -0.5), y, -1.0) * y), z, -t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -4.8e+65)
		tmp = t_1;
	elseif (x <= 2.8e+102)
		tmp = fma(Float64(fma(fma(-0.3333333333333333, y, -0.5), y, -1.0) * y), z, Float64(-t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.8e+65], t$95$1, If[LessEqual[x, 2.8e+102], N[(N[(N[(N[(-0.3333333333333333 * y + -0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y), $MachinePrecision] * z + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.8000000000000003e65 or 2.80000000000000018e102 < x

   1. Initial program 98.0%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot x} \]

      2. remove-double-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)} \cdot x \]

      3. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \log y}\right)\right) \cdot x \]

      4. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot \log y\right)\right)} \cdot x \]

      5. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}\right) \cdot x \]

      6. log-rec N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\log \left(\frac{1}{y}\right)}\right) \cdot x \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \log \left(\frac{1}{y}\right)\right) \cdot x} \]

      8. log-rec N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}\right) \cdot x \]

      9. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \color{blue}{\left(-1 \cdot \log y\right)}\right) \cdot x \]

      10. mul-1-neg N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \log y\right)\right)} \cdot x \]

      11. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}\right)\right) \cdot x \]

      12. remove-double-neg N/A

          \[\leadsto \color{blue}{\log y} \cdot x \]

      13. lower-log.f64 81.8

          \[\leadsto \color{blue}{\log y} \cdot x \]

   5. Applied rewrites 81.8%

      \[\leadsto \color{blue}{\log y \cdot x} \]

   if -4.8000000000000003e65 < x < 2.80000000000000018e102

   1. Initial program 78.9%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right) - t} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right) + \left(\mathsf{neg}\left(t\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log \left(1 - y\right) \cdot z} + \left(\mathsf{neg}\left(t\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(1 - y\right), z, \mathsf{neg}\left(t\right)\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)}, z, \mathsf{neg}\left(t\right)\right) \]

      5. lower-log1p.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)}, z, \mathsf{neg}\left(t\right)\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{-y}\right), z, \mathsf{neg}\left(t\right)\right) \]

      7. lower-neg.f64 81.8

         \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{-t}\right) \]

   5. Applied rewrites 81.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right), z, -t\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 81.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right), y, -1\right) \cdot y, z, -t\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+65}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right), y, -1\right) \cdot y, z, -t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-y, z, \mathsf{fma}\left(\log y, x, -t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma (- y) z (fma (log y) x (- t))))

C

double code(double x, double y, double z, double t) {
	return fma(-y, z, fma(log(y), x, -t));
}

Julia

function code(x, y, z, t)
	return fma(Float64(-y), z, fma(log(y), x, Float64(-t)))
end

Wolfram

code[x_, y_, z_, t_] := N[((-y) * z + N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.4%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t} \]

   2. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right)} - t \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(z \cdot \log \left(1 - y\right) + x \cdot \log y\right)} - t \]

   4. associate--l+ N/A

      \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right) + \left(x \cdot \log y - t\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right)} + \left(x \cdot \log y - t\right) \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\log \left(1 - y\right) \cdot z} + \left(x \cdot \log y - t\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(1 - y\right), z, x \cdot \log y - t\right)} \]

   8. lift-log.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\log \left(1 - y\right)}, z, x \cdot \log y - t\right) \]

   9. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\log \color{blue}{\left(1 - y\right)}, z, x \cdot \log y - t\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)}, z, x \cdot \log y - t\right) \]

   11. lower-log1p.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)}, z, x \cdot \log y - t\right) \]

   12. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{-y}\right), z, x \cdot \log y - t\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{x \cdot \log y + \left(\mathsf{neg}\left(t\right)\right)}\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{x \cdot \log y} + \left(\mathsf{neg}\left(t\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{\log y \cdot x} + \left(\mathsf{neg}\left(t\right)\right)\right) \]

   16. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{\mathsf{fma}\left(\log y, x, \mathsf{neg}\left(t\right)\right)}\right) \]

   17. lower-neg.f64 99.8

       \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, \color{blue}{-t}\right)\right) \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)} \]

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{-1 \cdot y}, z, \mathsf{fma}\left(\log y, x, -t\right)\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(y\right)}, z, \mathsf{fma}\left(\log y, x, -t\right)\right) \]

   2. lower-neg.f64 99.2

      \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, z, \mathsf{fma}\left(\log y, x, -t\right)\right) \]

7. Applied rewrites 99.2%

   \[\leadsto \mathsf{fma}\left(\color{blue}{-y}, z, \mathsf{fma}\left(\log y, x, -t\right)\right) \]
8. Add Preprocessing

Alternative 7: 99.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\log y, x, -\mathsf{fma}\left(z, y, t\right)\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma (log y) x (- (fma z y t))))

C

double code(double x, double y, double z, double t) {
	return fma(log(y), x, -fma(z, y, t));
}

Julia

function code(x, y, z, t)
	return fma(log(y), x, Float64(-fma(z, y, t)))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Log[y], $MachinePrecision] * x + (-N[(z * y + t), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 86.4%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right) - t} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot \log y + -1 \cdot \left(y \cdot z\right)\right)} - t \]

   2. associate--l+ N/A

      \[\leadsto \color{blue}{x \cdot \log y + \left(-1 \cdot \left(y \cdot z\right) - t\right)} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\log y \cdot x} + \left(-1 \cdot \left(y \cdot z\right) - t\right) \]

   4. remove-double-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)} \cdot x + \left(-1 \cdot \left(y \cdot z\right) - t\right) \]

   5. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \log y}\right)\right) \cdot x + \left(-1 \cdot \left(y \cdot z\right) - t\right) \]

   6. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \left(-1 \cdot \log y\right)\right)} \cdot x + \left(-1 \cdot \left(y \cdot z\right) - t\right) \]

   7. mul-1-neg N/A

      \[\leadsto \left(-1 \cdot \color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}\right) \cdot x + \left(-1 \cdot \left(y \cdot z\right) - t\right) \]

   8. log-rec N/A

      \[\leadsto \left(-1 \cdot \color{blue}{\log \left(\frac{1}{y}\right)}\right) \cdot x + \left(-1 \cdot \left(y \cdot z\right) - t\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \log \left(\frac{1}{y}\right), x, -1 \cdot \left(y \cdot z\right) - t\right)} \]

   10. log-rec N/A

       \[\leadsto \mathsf{fma}\left(-1 \cdot \color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}, x, -1 \cdot \left(y \cdot z\right) - t\right) \]

   11. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(-1 \cdot \color{blue}{\left(-1 \cdot \log y\right)}, x, -1 \cdot \left(y \cdot z\right) - t\right) \]

   12. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(-1 \cdot \log y\right)}, x, -1 \cdot \left(y \cdot z\right) - t\right) \]

   13. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log y\right)\right)}\right), x, -1 \cdot \left(y \cdot z\right) - t\right) \]

   14. remove-double-neg N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, x, -1 \cdot \left(y \cdot z\right) - t\right) \]

   15. lower-log.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\log y}, x, -1 \cdot \left(y \cdot z\right) - t\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{-1 \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(t\right)\right)}\right) \]

   17. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\mathsf{neg}\left(y \cdot z\right)\right)} + \left(\mathsf{neg}\left(t\right)\right)\right) \]

   18. distribute-neg-out N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\mathsf{neg}\left(\left(y \cdot z + t\right)\right)}\right) \]

   19. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{-\left(y \cdot z + t\right)}\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\log y, x, -\left(\color{blue}{z \cdot y} + t\right)\right) \]

   21. lower-fma.f64 99.2

       \[\leadsto \mathsf{fma}\left(\log y, x, -\color{blue}{\mathsf{fma}\left(z, y, t\right)}\right) \]

5. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, -\mathsf{fma}\left(z, y, t\right)\right)} \]
6. Add Preprocessing

Alternative 8: 56.9% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right), y, -1\right) \cdot y, z, -t\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma (* (fma (fma -0.3333333333333333 y -0.5) y -1.0) y) z (- t)))

C

double code(double x, double y, double z, double t) {
	return fma((fma(fma(-0.3333333333333333, y, -0.5), y, -1.0) * y), z, -t);
}

Julia

function code(x, y, z, t)
	return fma(Float64(fma(fma(-0.3333333333333333, y, -0.5), y, -1.0) * y), z, Float64(-t))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(-0.3333333333333333 * y + -0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y), $MachinePrecision] * z + (-t)), $MachinePrecision]

Derivation

1. Initial program 86.4%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right) - t} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right) + \left(\mathsf{neg}\left(t\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\log \left(1 - y\right) \cdot z} + \left(\mathsf{neg}\left(t\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(1 - y\right), z, \mathsf{neg}\left(t\right)\right)} \]

   4. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)}, z, \mathsf{neg}\left(t\right)\right) \]

   5. lower-log1p.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)}, z, \mathsf{neg}\left(t\right)\right) \]

   6. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{-y}\right), z, \mathsf{neg}\left(t\right)\right) \]

   7. lower-neg.f64 57.4

      \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{-t}\right) \]

5. Applied rewrites 57.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \mathsf{fma}\left(y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right), z, -t\right) \]
7. Step-by-step derivation

8. Applied rewrites 57.4%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right), y, -1\right) \cdot y, z, -t\right) \]
9. Add Preprocessing

Alternative 9: 47.3% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-11}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-55}:\\ \;\;\;\;-z \cdot y\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2e-11) (- t) (if (<= t 3.1e-55) (- (* z y)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2e-11) {
		tmp = -t;
	} else if (t <= 3.1e-55) {
		tmp = -(z * y);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2d-11)) then
        tmp = -t
    else if (t <= 3.1d-55) then
        tmp = -(z * y)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2e-11) {
		tmp = -t;
	} else if (t <= 3.1e-55) {
		tmp = -(z * y);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2e-11:
		tmp = -t
	elif t <= 3.1e-55:
		tmp = -(z * y)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2e-11)
		tmp = Float64(-t);
	elseif (t <= 3.1e-55)
		tmp = Float64(-Float64(z * y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2e-11)
		tmp = -t;
	elseif (t <= 3.1e-55)
		tmp = -(z * y);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2e-11], (-t), If[LessEqual[t, 3.1e-55], (-N[(z * y), $MachinePrecision]), (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -1.99999999999999988e-11 or 3.09999999999999997e-55 < t

   1. Initial program 96.6%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{-1 \cdot t} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(t\right)} \]

      2. lower-neg.f64 67.8

         \[\leadsto \color{blue}{-t} \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{-t} \]

   if -1.99999999999999988e-11 < t < 3.09999999999999997e-55

   1. Initial program 73.2%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right) - t} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right) + \left(\mathsf{neg}\left(t\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\log \left(1 - y\right) \cdot z} + \left(\mathsf{neg}\left(t\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(1 - y\right), z, \mathsf{neg}\left(t\right)\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)}, z, \mathsf{neg}\left(t\right)\right) \]

      5. lower-log1p.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)}, z, \mathsf{neg}\left(t\right)\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{-y}\right), z, \mathsf{neg}\left(t\right)\right) \]

      7. lower-neg.f64 39.4

         \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{-t}\right) \]

   5. Applied rewrites 39.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \left(y \cdot z\right) - \color{blue}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.7%

      \[\leadsto -\mathsf{fma}\left(z, y, t\right) \]

   9. Taylor expanded in t around 0

      \[\leadsto -y \cdot z \]
   10. Step-by-step derivation

   11. Applied rewrites 28.2%

       \[\leadsto -z \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 56.8% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot y, z, -t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma (* (fma -0.5 y -1.0) y) z (- t)))

C

double code(double x, double y, double z, double t) {
	return fma((fma(-0.5, y, -1.0) * y), z, -t);
}

Julia

function code(x, y, z, t)
	return fma(Float64(fma(-0.5, y, -1.0) * y), z, Float64(-t))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(-0.5 * y + -1.0), $MachinePrecision] * y), $MachinePrecision] * z + (-t)), $MachinePrecision]

Derivation

1. Initial program 86.4%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right) - t} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right) + \left(\mathsf{neg}\left(t\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\log \left(1 - y\right) \cdot z} + \left(\mathsf{neg}\left(t\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(1 - y\right), z, \mathsf{neg}\left(t\right)\right)} \]

   4. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)}, z, \mathsf{neg}\left(t\right)\right) \]

   5. lower-log1p.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)}, z, \mathsf{neg}\left(t\right)\right) \]

   6. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{-y}\right), z, \mathsf{neg}\left(t\right)\right) \]

   7. lower-neg.f64 57.4

      \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{-t}\right) \]

5. Applied rewrites 57.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto \mathsf{fma}\left(y \cdot \left(\frac{-1}{2} \cdot y - 1\right), z, -t\right) \]
7. Step-by-step derivation

8. Applied rewrites 57.4%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot y, z, -t\right) \]
9. Add Preprocessing

Alternative 11: 56.5% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ -\mathsf{fma}\left(z, y, t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (fma z y t)))

C

double code(double x, double y, double z, double t) {
	return -fma(z, y, t);
}

Julia

function code(x, y, z, t)
	return Float64(-fma(z, y, t))
end

Wolfram

code[x_, y_, z_, t_] := (-N[(z * y + t), $MachinePrecision])

Derivation

1. Initial program 86.4%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right) - t} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right) + \left(\mathsf{neg}\left(t\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\log \left(1 - y\right) \cdot z} + \left(\mathsf{neg}\left(t\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(1 - y\right), z, \mathsf{neg}\left(t\right)\right)} \]

   4. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)}, z, \mathsf{neg}\left(t\right)\right) \]

   5. lower-log1p.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)}, z, \mathsf{neg}\left(t\right)\right) \]

   6. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(\color{blue}{-y}\right), z, \mathsf{neg}\left(t\right)\right) \]

   7. lower-neg.f64 57.4

      \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{-t}\right) \]

5. Applied rewrites 57.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)} \]

6. Taylor expanded in y around 0

   \[\leadsto -1 \cdot \left(y \cdot z\right) - \color{blue}{t} \]
7. Step-by-step derivation

8. Applied rewrites 57.0%

   \[\leadsto -\mathsf{fma}\left(z, y, t\right) \]
9. Add Preprocessing

Alternative 12: 42.2% accurate, 73.3× speedup

Math

\[\begin{array}{l} \\ -t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- t))

C

double code(double x, double y, double z, double t) {
	return -t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = -t
end function

Java

public static double code(double x, double y, double z, double t) {
	return -t;
}

Python

def code(x, y, z, t):
	return -t

Julia

function code(x, y, z, t)
	return Float64(-t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = -t;
end

Wolfram

code[x_, y_, z_, t_] := (-t)

Derivation

1. Initial program 86.4%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{-1 \cdot t} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(t\right)} \]

   2. lower-neg.f64 44.0

      \[\leadsto \color{blue}{-t} \]

5. Applied rewrites 44.0%

   \[\leadsto \color{blue}{-t} \]
6. Add Preprocessing

Alternative 13: 2.3% accurate, 220.0× speedup

Math

\[\begin{array}{l} \\ t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 t)

C

double code(double x, double y, double z, double t) {
	return t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = t
end function

Java

public static double code(double x, double y, double z, double t) {
	return t;
}

Python

def code(x, y, z, t):
	return t

Julia

function code(x, y, z, t)
	return t
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = t;
end

Wolfram

code[x_, y_, z_, t_] := t

Derivation

1. Initial program 86.4%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in t around inf

   \[\leadsto \color{blue}{-1 \cdot t} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(t\right)} \]

   2. lower-neg.f64 44.0

      \[\leadsto \color{blue}{-t} \]

5. Applied rewrites 44.0%

   \[\leadsto \color{blue}{-t} \]
6. Step-by-step derivation

7. Applied rewrites 23.9%

   \[\leadsto \frac{0 - t \cdot t}{\color{blue}{0 + t}} \]
8. Step-by-step derivation

9. Applied rewrites 2.3%

   \[\leadsto t \]
10. Add Preprocessing

Developer Target 1: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (*
   (- z)
   (+
    (+ (* 0.5 (* y y)) y)
    (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y)))))
  (- t (* x (log y)))))

C

double code(double x, double y, double z, double t) {
	return (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * log(y)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (-z * (((0.5d0 * (y * y)) + y) + ((0.3333333333333333d0 / (1.0d0 * (1.0d0 * 1.0d0))) * (y * (y * y))))) - (t - (x * log(y)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * Math.log(y)));
}

Python

def code(x, y, z, t):
	return (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * math.log(y)))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(-z) * Float64(Float64(Float64(0.5 * Float64(y * y)) + y) + Float64(Float64(0.3333333333333333 / Float64(1.0 * Float64(1.0 * 1.0))) * Float64(y * Float64(y * y))))) - Float64(t - Float64(x * log(y))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * log(y)));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[((-z) * N[(N[(N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + N[(N[(0.3333333333333333 / N[(1.0 * N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t - N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024254 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* (- z) (+ (+ (* 1/2 (* y y)) y) (* (/ 1/3 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y)))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))